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update fermi example

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NunoSempere 2024-01-20 19:48:55 +01:00
parent 1d89eb6231
commit 4b158c95df
3 changed files with 61 additions and 104 deletions
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examples/core/06_dissolving_fermi_paradox/example
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examples/core/06_dissolving_fermi_paradox/example.c
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@ -4,6 +4,8 @@
#include <stdio.h> #include <stdio.h>
#include <stdlib.h> #include <stdlib.h>
#define VERBOSE 0
double sample_loguniform(double a, double b, uint64_t* seed) double sample_loguniform(double a, double b, uint64_t* seed)
{ {
return exp(sample_uniform(log(a), log(b), seed)); return exp(sample_uniform(log(a), log(b), seed));
@ -18,6 +20,7 @@ int main()
uint64_t* seed = malloc(sizeof(uint64_t)); uint64_t* seed = malloc(sizeof(uint64_t));
*seed = UINT64_MAX / 64; // xorshift can't start with a seed of 0 *seed = UINT64_MAX / 64; // xorshift can't start with a seed of 0
// Do this naïvely, without worrying that much about numerical precision
double sample_fermi_naive(uint64_t * seed) double sample_fermi_naive(uint64_t * seed)
{ {
double rate_of_star_formation = sample_loguniform(1, 100, seed); double rate_of_star_formation = sample_loguniform(1, 100, seed);
@ -31,14 +34,14 @@ int main()
double fraction_of_intelligent_planets_which_are_detectable_as_such = sample_loguniform(0.01, 1, seed); double fraction_of_intelligent_planets_which_are_detectable_as_such = sample_loguniform(0.01, 1, seed);
double longevity_of_detectable_civilizations = sample_loguniform(100, 10000000000, seed); double longevity_of_detectable_civilizations = sample_loguniform(100, 10000000000, seed);
// printf(" rate_of_star_formation = %lf\n", rate_of_star_formation); if(VERBOSE) printf(" rate_of_star_formation = %lf\n", rate_of_star_formation);
// printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets); if(VERBOSE) printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets);
// printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system); if(VERBOSE) printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system);
// printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets); if(VERBOSE) printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets);
// printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears); if(VERBOSE) printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears);
// printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears); if(VERBOSE) printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears);
// printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such); if(VERBOSE) printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such);
// printf(" longevity_of_detectable_civilizations = %lf\n", longevity_of_detectable_civilizations); if(VERBOSE) printf(" longevity_of_detectable_civilizations = %lf\n", longevity_of_detectable_civilizations);
// Expected number of civilizations in the Milky way; // Expected number of civilizations in the Milky way;
// see footnote 3 (p. 5) // see footnote 3 (p. 5)
@ -47,7 +50,7 @@ int main()
return n; return n;
} }
double sample_fermi_paradox_naive(uint64_t * seed) double sample_are_we_alone_naive(uint64_t * seed)
{ {
double n = sample_fermi_naive(seed); double n = sample_fermi_naive(seed);
return ((n > 1) ? 1 : 0); return ((n > 1) ? 1 : 0);
@ -56,13 +59,14 @@ int main()
double n = 1000000; double n = 1000000;
double naive_fermi_proportion = 0; double naive_fermi_proportion = 0;
for (int i = 0; i < n; i++) { for (int i = 0; i < n; i++) {
double result = sample_fermi_paradox_naive(seed); double result = sample_are_we_alone_naive(seed);
// printf("result: %lf\n", result); if(VERBOSE) printf("result: %lf\n", result);
naive_fermi_proportion += result; naive_fermi_proportion += result;
} }
printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion / n); printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion / n);
// Thinking in log space // Thinking in log space
// Taking care of numerical precision
double sample_fermi_logspace(uint64_t * seed) double sample_fermi_logspace(uint64_t * seed)
{ {
double log_rate_of_star_formation = sample_uniform(log(1), log(100), seed); double log_rate_of_star_formation = sample_uniform(log(1), log(100), seed);
@ -72,64 +76,63 @@ int main()
double log_fraction_of_intelligent_planets_which_are_detectable_as_such = sample_uniform(log(0.01), log(1), seed); double log_fraction_of_intelligent_planets_which_are_detectable_as_such = sample_uniform(log(0.01), log(1), seed);
double log_longevity_of_detectable_civilizations = sample_uniform(log(100), log(10000000000), seed); double log_longevity_of_detectable_civilizations = sample_uniform(log(100), log(10000000000), seed);
// printf(" log_rate_of_star_formation = %lf\n", log_rate_of_star_formation); if(VERBOSE) printf(" log_rate_of_star_formation = %lf\n", log_rate_of_star_formation);
// printf(" log_fraction_of_stars_with_planets = %lf\n", log_fraction_of_stars_with_planets); if(VERBOSE) printf(" log_fraction_of_stars_with_planets = %lf\n", log_fraction_of_stars_with_planets);
// printf(" log_number_of_habitable_planets_per_star_system = %lf\n", log_number_of_habitable_planets_per_star_system); if(VERBOSE) printf(" log_number_of_habitable_planets_per_star_system = %lf\n", log_number_of_habitable_planets_per_star_system);
// printf(" log_fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", log_fraction_of_planets_with_life_in_which_intelligent_life_appears); if(VERBOSE) printf(" log_fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", log_fraction_of_planets_with_life_in_which_intelligent_life_appears);
// printf(" log_fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", log_fraction_of_intelligent_planets_which_are_detectable_as_such); if(VERBOSE) printf(" log_fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", log_fraction_of_intelligent_planets_which_are_detectable_as_such);
// printf(" log_longevity_of_detectable_civilizations = %lf\n", log_longevity_of_detectable_civilizations); if(VERBOSE) printf(" log_longevity_of_detectable_civilizations = %lf\n", log_longevity_of_detectable_civilizations);
double log_n1 = log_rate_of_star_formation + log_fraction_of_stars_with_planets + log_number_of_habitable_planets_per_star_system + log_fraction_of_planets_with_life_in_which_intelligent_life_appears + log_fraction_of_intelligent_planets_which_are_detectable_as_such + log_longevity_of_detectable_civilizations; double log_n1 = log_rate_of_star_formation + log_fraction_of_stars_with_planets + log_number_of_habitable_planets_per_star_system + log_fraction_of_planets_with_life_in_which_intelligent_life_appears + log_fraction_of_intelligent_planets_which_are_detectable_as_such + log_longevity_of_detectable_civilizations;
// printf("first part of calculation: %lf\n", log_n1); if(VERBOSE) printf("first part of calculation: %lf\n", log_n1);
/* Consider fraction_of_habitable_planets_in_which_any_life_appears separately. /*
Imprecisely, we could do: Consider:
double rate_of_life_formation_in_habitable_planets = sample_lognormal(1, 50, seed);
double fraction_of_habitable_planets_in_which_any_life_appears = 1- exp(-rate_of_life_formation_in_habitable_planets);
double log_fraction_of_habitable_planets_in_which_any_life_appears = log(1-fraction_of_habitable_planets_in_which_any_life_appears);
double n = exp(log_n1) * fraction_of_habitable_planets_in_which_any_life_appears;
// or:
double n2 = exp(log_n1 + log(fraction_of_habitable_planets_in_which_any_life_appears))
However, we lose all precision here.
Now, say
a = underlying normal a = underlying normal
b = rate_of_life_formation_in_habitable_planets = exp(underlying normal) b = rate_of_life_formation_in_habitable_planets = exp(underlying normal) = exp(a)
c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears
d = log(c) d = log(c)
Now, is there some way we can d more efficiently/precisely? Now, is there some way we can get d more efficiently/precisely?
Turns out there is!
Turns out there is!
Looking at the Taylor expansion for c = 1 - exp(-b), it's b - b^2/2 + b^3/6 - x^b/24, etc. Looking at the Taylor expansion for c = 1 - exp(-b), it's b - b^2/2 + b^3/6 - x^b/24, etc.
// https://www.wolframalpha.com/input?i=1-exp%28-x%29 <https://www.wolframalpha.com/input?i=1-exp%28-x%29>
When b ~ 0 (as is often the case), this is close to b. When b ~ 0 (as is often the case), this is close to b.
But now, if b ~ 0 But now, if b ~ 0
c ~ b c ~ b
and d = log(c) ~ log(b) = log(exp(a)) = a and d = log(c) ~ log(b) = log(exp(a)) = a
Now, we could play around with estimating errors,
and indeed if we want b^2/2 = exp(a)^2/2 < 10^(-n), i.e., to have n decimal digits of precision,
we could compute this as e.g., a < (nlog(10) + log(2))/2
so for example if we want ten digits of precision, that's a < -11
But more empirically, the two numbers do become really close around 11 or so, and at 38 that calculation results in a -inf (so probably an overflow.)
So we should be using that formula for somewhere between -38 << a < -11
I chose -16 for the sake of it after playing with:
<https://www.wolframalpha.com/input?i=log%281-exp%28-exp%28-16%29%29%29>
*/ */
double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed); double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed);
// printf("log_rate_of_life_formation_in_habitable_planets: %lf\n", log_rate_of_life_formation_in_habitable_planets); if(VERBOSE) printf("log_rate_of_life_formation_in_habitable_planets: %lf\n", log_rate_of_life_formation_in_habitable_planets);
double log_fraction_of_habitable_planets_in_which_any_life_appears; double log_fraction_of_habitable_planets_in_which_any_life_appears;
if (log_rate_of_life_formation_in_habitable_planets < -32) { if (log_rate_of_life_formation_in_habitable_planets < -16) {
log_fraction_of_habitable_planets_in_which_any_life_appears = log_rate_of_life_formation_in_habitable_planets; log_fraction_of_habitable_planets_in_which_any_life_appears = log_rate_of_life_formation_in_habitable_planets;
} else { } else {
double rate_of_life_formation_in_habitable_planets = exp(log_rate_of_life_formation_in_habitable_planets); double rate_of_life_formation_in_habitable_planets = exp(log_rate_of_life_formation_in_habitable_planets);
double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets); double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
log_fraction_of_habitable_planets_in_which_any_life_appears = log(fraction_of_habitable_planets_in_which_any_life_appears); log_fraction_of_habitable_planets_in_which_any_life_appears = log(fraction_of_habitable_planets_in_which_any_life_appears);
} }
// printf(" log_fraction_of_habitable_planets_in_which_any_life_appears: %lf\n", log_fraction_of_habitable_planets_in_which_any_life_appears); if(VERBOSE) printf(" log_fraction_of_habitable_planets_in_which_any_life_appears: %lf\n", log_fraction_of_habitable_planets_in_which_any_life_appears);
double log_n = log_n1 + log_fraction_of_habitable_planets_in_which_any_life_appears; double log_n = log_n1 + log_fraction_of_habitable_planets_in_which_any_life_appears;
return log_n; return log_n;
} }
double sample_fermi_paradox_logspace(uint64_t * seed) double sample_are_we_alone_logspace(uint64_t * seed)
{ {
double n = sample_fermi_logspace(seed); double n = sample_fermi_logspace(seed);
return ((n > 0) ? 1 : 0); return ((n > 0) ? 1 : 0);
@ -137,11 +140,21 @@ int main()
double logspace_fermi_proportion = 0; double logspace_fermi_proportion = 0;
for (int i = 0; i < n; i++) { for (int i = 0; i < n; i++) {
double result = sample_fermi_paradox_logspace(seed); double result = sample_are_we_alone_logspace(seed);
// printf("result: %lf\n", result); if(VERBOSE) printf("result: %lf\n", result);
logspace_fermi_proportion += result; logspace_fermi_proportion += result;
} }
printf("Using more accurate logspace computations, %% that we are not alone: %lf\n", logspace_fermi_proportion / n); printf("Using more accurate logspace computations, %% that we are not alone: %lf\n", logspace_fermi_proportion / n);
free(seed); free(seed);
/*
double invert(double x){
return log(1-exp(-exp(-x)));
}
for(int i=0; i<64; i++){
double j = i;
printf("for %lf, log(1-exp(-exp(-x))) is calculated as... %lf\n", j, invert(j));
}
*/
} }

56
examples/core/06_dissolving_fermi_paradox/makefile
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@ -1,56 +0,0 @@
# Interface:
# make
# make build
# make format
# make run
# Compiler
CC=gcc
# CC=tcc # <= faster compilation
# Main file
SRC=scratchpad.c ../squiggle.c ../squiggle_more.c
OUTPUT=scratchpad
## Dependencies
MATH=-lm
## Flags
DEBUG= #'-g'
STANDARD=-std=c99
WARNINGS=-Wall
OPTIMIZED=-O3 #-Ofast
# OPENMP=-fopenmp
## Formatter
STYLE_BLUEPRINT=webkit
FORMATTER=clang-format -i -style=$(STYLE_BLUEPRINT)
## make build
build: $(SRC)
$(CC) $(OPTIMIZED) $(DEBUG) $(SRC) $(MATH) -o $(OUTPUT)
format: $(SRC)
$(FORMATTER) $(SRC)
run: $(SRC) $(OUTPUT)
./$(OUTPUT)
verify: $(SRC) $(OUTPUT)
./$(OUTPUT) | grep "NOT passed" -A 2 --group-separator='' || true
time-linux:
@echo "Requires /bin/time, found on GNU/Linux systems" && echo
@echo "Running 100x and taking avg time $(OUTPUT)"
@t=$$(/usr/bin/time -f "%e" -p bash -c 'for i in {1..100}; do $(OUTPUT); done' 2>&1 >/dev/null | grep real | awk '{print $$2}' ); echo "scale=2; 1000 * $$t / 100" | bc | sed "s|^|Time using 1 thread: |" | sed 's|$$|ms|' && echo
## Profiling
profile-linux:
echo "Requires perf, which depends on the kernel version, and might be in linux-tools package or similar"
echo "Must be run as sudo"
$(CC) $(SRC) $(MATH) -o $(OUTPUT)
sudo perf record ./$(OUTPUT)
sudo perf report
rm perf.data